A new method for modelling the stochastic differential equations

Abstract

This study presents a novel approach to estimate the probability density function of solutions to stochastic differential equations using generalized entropy optimization methods. Unlike traditional methods such as the Fokker–Planck–Kolmogorov equation, the proposed generalized entropy optimization methods framework accommodates cases where the distribution of the solution deviates from standard statistical forms. The method integrates the Euler–Maruyama scheme to generate multiple trajectories, producing random variables $\hat{X}(t)$ for each time $t$. The performance of method is evaluated through a comprehensive simulation study, in which it is compared with existing techniques under various parameter settings. Both generalized MaxEnt and MinxEnt distributions are applied, with results indicating that generalized MinxEnt distributions offer superior adaptability and accuracy. Visual and statistical comparisons confirm the theoretical validity and practical efficiency of the method. This framework not only provides a flexible alternative for probability density function estimation in stochastic differential equation modeling but also opens pathways for applications in fuzzy stochastic differential equation systems.

Keywords

• Euler-Maruyama method
• generalized entropy optimization methods
• stochastic differential equations
• stochastic process

Supporting Institution

This study is supported by the Eskisehir Technical University Scientific Research Projects Commission under grant No. 20DRP046.

Project Number

This study is supported by the Eskisehir Technical University Scientific Research Projects Commission under grant No. 20DRP046

Ethical Statement

The author declares that has no conflict of interest.

Thanks

We would like to thank Prof. Dr. Aladdin SHAMILOV for the continuous support of knowledge and theoretical support

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